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KEEPIN’ IT REAL
THE REAL NUMBER SYSTEM
UNIT 2
• I can classify real numbers
• I can plot real numbers on the number line and order real numbers.
• I can interpret change as positive or negative numbers.
• I can find the opposite and the absolute value of a real number.
WITH A PARTNER, LIST EXAMPLES OF AS
MANY DIFFERENT TYPES OF NUMBERS AS
YOU CAN THINK OF.
REAL NUMBERS
Irrational Numbers
Rational Numbers
¼
½
1/3
Integers
Natural
Numbers:
1,2,3,…
√2
-2
-1
0
¼
13/8
1.673
√3
• Integers: Positive and
Negative Natural
Numbers (and 0)
• Rational Numbers:
Fractions of Integers,
Decimals with repeating
patterns.
• Irrational Numbers:
Have infinite decimal
expansions with no
patterns
REAL NUMBERS
Irrational Numbers
Rational Numbers
¼
½
1/3
Integers
Natural
Numbers:
1,2,3,…
√2
-2
-1
0
¼
13/8
1.673
√3
• Natural numbers are
integers.
• Integers are rational
numbers:
6 = 6/1
• Rational numbers are
NOT irrational.
NATURAL NUMBER? INTEGER? RATIONAL
NUMBER? IRRATIONAL NUMBER?
•
1.
2.
3.
4.
5.
6.
7.
List all that apply:
51351390
-456.14589
3½
¼
-14/5
√13
√9
CREATE A REAL NUMBER SYSTEM
The Real Number Line
The negative real numbers are the coordinates
of points to the left of the origin 0.
The real number zero is the coordinate of the
origin O.
The positive real numbers are the coordinates of
points to the right of the origin O.
THINK OF I-10
• Exit 194 is 0, the
origin.
• The “positive”
direction is to the
right.
• The “negative”
direction is to the
left.
THINK OF I-10
• Imagine you are trying
to go as far to the
“right” as possible.
• What is closer to your
goal: 5 miles from the
exit or -3 miles from
the exit?
THINK OF I-10
• Imagine you are trying
to go as far to the
“right” as possible.
• What is closer to your
goal: -7 miles from the
exit or -3 miles from
the exit?
ORDERING REAL NUMBERS
• The symbols:
1. a < b (a is less than b)
2. a > b (a is greater than
b)
3. a = b (a is equal to b)
• The new rules:
1. If a is negative and b is
postive: a < b
2. If a and b are positive
and a > b, than – a < b.
• Examples:
1. -3 < 5
2. -7 < -3
DO YOU KNOW HOW?
• On your number line, plot:
-7, 9, -3/2, 2.7, 5.9, and ¼
• Which is greater, -143 or 12?
• Which is greater, -41 or -1?
• Which is greater, 0 or 5?
• Which is greater, 0 or -5?
WHAT DO POSITIVE AND
NEGATIVE NUMBERS MEAN?
To which of the following words describing change
would you associate with positive numbers?
Which with negative numbers?
decrease
below sea level
+
credit
+
surplus
+
gain
loss
+
increase
Can you think of any more?
deficit
debit
+
above sea level
USE AN INTEGER TO DESCRIBE THE
FOLLOWING:
•
•
•
•
•
•
•
Kalamazoo is 780 feet above sea level.
I lost $5 betting at the track.
The temperature decreased by 7 degrees.
I dove 20 feet below sea level.
I made $143 on that stock!
The temperature warmed up by 3 degrees.
Illegal formation: 10 yard penalty!
OPPOSITES
•
To find the opposite of a (nonzero) real number, change its sign.
•
The opposite is equally far from the origin, but in the “opposite” direction.
OPPOSITES
•
To find the opposite of a (nonzero) real number, change its sign.
•
Find the opposite of:
1.
679
2.
-34
3.
-13
4.
¼
DISTANCE AND ABSOLUTE
VALUE
• A distance is never negative
• The absolute value of a number is its distance
from the origin on the number line.
NUMBER LINE
• How far is 3 from
zero?
• How far is -3 from
zero?
|X| THE ABSOLUTE VALUE OF
X
• |3| asks how far from zero is 3?
• |-5| asks how far from zero is -5?
ABSOLUTE VALUE
• Always gives a positive answer or zero.
• If there is arithmetic inside the absolute value
symbol do that first, then take the absolute value
of the answer.
• Real numbers include natural numbers, whole numbers,
integers, rational numbers, and irrational numbers.
• Real numbers can be laid out along a number line.
• Positive numbers > Negative Numbers
• Negative numbers are ordered in reverse
• Positive and negative numbers can describe change.
• Changing the sign of a real number gives its opposite.
• Absolute value is like distance, sign is like a direction.